## Probability in the North East day

#### 8 May 2019

University of York.

Organizer: Stephen Connor.

These people attended the meeting.

## Programme

12:30–13:10
Lunch
13:10–14:00
Aihua Zhang (University of Leicester)
In this paper we extend the consumption-investment life cycle model for an uncertain-lived agent, proposed by Richard (1974), to allow for flexible labor supply. We study the consumption, labor supply and portfolio decisions of an agent facing age-dependent mortality risk, as presented by UK actuarial life tables spanning the time period from 1951-2060 (including mortality forecasts). We find that historical changes in mortality produces significant changes in portfolio investment (more risk taking), labour (decrease of hours) and consumption level (shift to higher level) contributing up to 5% to GDP growth for the last three decades.
14.00–14.50
Elena Issoglio (University of Leeds)
This talk focuses on a multidimensional SDE where the drift is an element of a fractional Sobolev space with negative order, hence a distribution. This SDE admits a unique weak solution in a suitable sense - this was proven in [Flandoli, Issoglio, Russo (2017)]. The aim here is to construct a numerical scheme to approximate this solution. One of the key problems is that the drift cannot be evaluated pointwise, hence we approximate it with suitable functions using Haar wavelets, and then apply (an extended version of) Euler-Maruyama scheme. We then show that the algorithm converges in law, and in the special 1-dimensional case we also get a rate of convergence.

This talk is based on a joint work with T. De Angelis and M. Germain.
14:50–15:20
Tea and coffee
15:20–16:10
Evita Nestoridi (University of Cambridge)
We study a natural random walk over the upper triangular matrices, with entries in $\mathbb{Z}/m\mathbb{Z}$, generated by steps which add or subtract row $i + 1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2m^2)$ which is optimal up to constants. This generalizes a result of Peres and Sly and answers a question of Stong and of Arias-Castro, Diaconis and Stanley.

This is joint work with Allan Sly.
16:10–17:00
Andrea Meireles Rodrigues (University of York)
This work studies optimal portfolio decisions when investors exhibit the reference-dependent preferences of Koszegi and Rabin, in a general complete market setting with a no-bankruptcy constraint. If loss aversion or gain-loss sensitivity are weak, a unique personal equilibrium arises that qualitatively resembles the classical utility maximiser. By contrast, strong loss aversion and gain-loss aversion can generate multiple (in fact, infinitely many) coexisting personal equilibria, whose distributions are a combination of continuous and discrete parts. We also investigate the sensitivity of the personal equilibria to market and preference parameters, as well as their limiting behaviours.